[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/schnirelmann-density-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki41\/schnirelmann-density-wikipedia\/","headline":"Schnirelmann density – Wikipedia","name":"Schnirelmann density – Wikipedia","description":"before-content-x4 In additive number theory, a way to measure how dense a sequence of numbers is after-content-x4 In additive number","datePublished":"2016-01-29","dateModified":"2016-01-29","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki41\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c028d50c3e1359bb874e4abebfd1678b578eeb5f","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c028d50c3e1359bb874e4abebfd1678b578eeb5f","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki41\/schnirelmann-density-wikipedia\/","about":["Wiki"],"wordCount":14839,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In additive number theory, a way to measure how dense a sequence of numbers is       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how “dense” the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.[1][2]Table of Contents       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Definition[edit]Properties[edit]Sensitivity[edit]Schnirelmann’s theorems[edit]Additive bases[edit]Mann’s theorem[edit]Waring’s problem[edit]Schnirelmann’s constant[edit]Essential components[edit]References[edit]Definition[edit]The Schnirelmann density of a set of natural numbers A is defined as\u03c3A=infnA(n)n,{displaystyle sigma A=inf _{n}{frac {A(n)}{n}},}where A(n) denotes the number of elements of A not exceeding n and inf is infimum.[3]The Schnirelmann density is well-defined even if the limit of A(n)\/n as n \u2192 \u221e fails to exist (see upper and lower asymptotic density).       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Properties[edit]By definition, 0 \u2264 A(n) \u2264 n and n \u03c3A \u2264 A(n) for all n, and therefore 0 \u2264 \u03c3A \u2264 1, and \u03c3A = 1 if and only if A = N. Furthermore,\u03c3A=0\u21d2\u2200\u03f5>0\u00a0\u2203n\u00a0A(n)k\u00a0k\u2209A\u21d2\u03c3A\u22641\u22121\/k{displaystyle forall k knotin ARightarrow sigma Aleq 1-1\/k}.In particular,1\u2209A\u21d2\u03c3A=0{displaystyle 1notin ARightarrow sigma A=0}and2\u2209A\u21d2\u03c3A\u226412.{displaystyle 2notin ARightarrow sigma Aleq {frac {1}{2}}.}Consequently, the Schnirelmann densities of the even numbers and the odd numbers, which one might expect to agree, are 0 and 1\/2 respectively. Schnirelmann and Yuri Linnik exploited this sensitivity as we shall see.Schnirelmann’s theorems[edit]If we set G2={k2}k=1\u221e{displaystyle {mathfrak {G}}^{2}={k^{2}}_{k=1}^{infty }}, then Lagrange’s four-square theorem can be restated as \u03c3(G2\u2295G2\u2295G2\u2295G2)=1{displaystyle sigma ({mathfrak {G}}^{2}oplus {mathfrak {G}}^{2}oplus {mathfrak {G}}^{2}oplus {mathfrak {G}}^{2})=1}. (Here the symbol A\u2295B{displaystyle Aoplus B} denotes the sumset of A\u222a{0}{displaystyle Acup {0}} and B\u222a{0}{displaystyle Bcup {0}}.) It is clear that \u03c3G2=0{displaystyle sigma {mathfrak {G}}^{2}=0}. In fact, we still have \u03c3(G2\u2295G2)=0{displaystyle sigma ({mathfrak {G}}^{2}oplus {mathfrak {G}}^{2})=0}, and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase. It actually is the case that \u03c3(G2\u2295G2\u2295G2)=5\/6{displaystyle sigma ({mathfrak {G}}^{2}oplus {mathfrak {G}}^{2}oplus {mathfrak {G}}^{2})=5\/6} and one sees that sumsetting G2{displaystyle {mathfrak {G}}^{2}} once again yields a more populous set, namely all of N{displaystyle mathbb {N} }. Schnirelmann further succeeded in developing these ideas into the following theorems, aiming towards Additive Number Theory, and proving them to be a novel resource (if not greatly powerful) to attack important problems, such as Waring’s problem and Goldbach’s conjecture.Theorem. Let A{displaystyle A} and B{displaystyle B} be subsets of N{displaystyle mathbb {N} }. Then\u03c3(A\u2295B)\u2265\u03c3A+\u03c3B\u2212\u03c3A\u22c5\u03c3B.{displaystyle sigma (Aoplus B)geq sigma A+sigma B-sigma Acdot sigma B.}Note that \u03c3A+\u03c3B\u2212\u03c3A\u22c5\u03c3B=1\u2212(1\u2212\u03c3A)(1\u2212\u03c3B){displaystyle sigma A+sigma B-sigma Acdot sigma B=1-(1-sigma A)(1-sigma B)}. Inductively, we have the following generalization.Corollary. Let Ai\u2286N{displaystyle A_{i}subseteq mathbb {N} } be a finite family of subsets of N{displaystyle mathbb {N} }. Then\u03c3(\u2a01iAi)\u22651\u2212\u220fi(1\u2212\u03c3Ai).{displaystyle sigma left(bigoplus _{i}A_{i}right)geq 1-prod _{i}left(1-sigma A_{i}right).}The theorem provides the first insights on how sumsets accumulate. It seems unfortunate that its conclusion stops short of showing \u03c3{displaystyle sigma } being superadditive. Yet, Schnirelmann provided us with the following results, which sufficed for most of his purpose.Theorem. Let A{displaystyle A} and B{displaystyle B} be subsets of N{displaystyle mathbb {N} }. If \u03c3A+\u03c3B\u22651{displaystyle sigma A+sigma Bgeq 1}, thenA\u2295B=N.{displaystyle Aoplus B=mathbb {N} .}Theorem. (Schnirelmann) Let A\u2286N{displaystyle Asubseteq mathbb {N} }. If "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki41\/schnirelmann-density-wikipedia\/#breadcrumbitem","name":"Schnirelmann density – Wikipedia"}}]}]